Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B

Average Error: 20.4 → 0.2

Time: 4.6s

Precision: binary64

Math

\[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -6500000000:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right) + \frac{y}{z} \cdot \left(0.07512208616047561 + \frac{-0.4046220386999212}{z}\right)\\ \mathbf{elif}\;z \leq 155000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \sqrt[3]{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)\right)}^{2}}, 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2.181088706546648 \cdot \frac{y}{{z}^{3}} + \left(\frac{y}{z} \cdot 0.07512208616047561 + \left(x + 0.0692910599291889 \cdot y\right)\right)\right) + \frac{y}{{z}^{2}} \cdot -0.4046220386999212\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -6500000000.0)
   (+
    (fma 0.0692910599291889 y x)
    (* (/ y z) (+ 0.07512208616047561 (/ -0.4046220386999212 z))))
   (if (<= z 155000.0)
     (fma
      y
      (/
       (fma
        z
        (*
         (cbrt (fma z 0.0692910599291889 0.4917317610505968))
         (cbrt (pow (fma z 0.0692910599291889 0.4917317610505968) 2.0)))
        0.279195317918525)
       (fma z (+ z 6.012459259764103) 3.350343815022304))
      x)
     (+
      (+
       (* 2.181088706546648 (/ y (pow z 3.0)))
       (+ (* (/ y z) 0.07512208616047561) (+ x (* 0.0692910599291889 y))))
      (* (/ y (pow z 2.0)) -0.4046220386999212)))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6500000000.0) {
		tmp = fma(0.0692910599291889, y, x) + ((y / z) * (0.07512208616047561 + (-0.4046220386999212 / z)));
	} else if (z <= 155000.0) {
		tmp = fma(y, (fma(z, (cbrt(fma(z, 0.0692910599291889, 0.4917317610505968)) * cbrt(pow(fma(z, 0.0692910599291889, 0.4917317610505968), 2.0))), 0.279195317918525) / fma(z, (z + 6.012459259764103), 3.350343815022304)), x);
	} else {
		tmp = ((2.181088706546648 * (y / pow(z, 3.0))) + (((y / z) * 0.07512208616047561) + (x + (0.0692910599291889 * y)))) + ((y / pow(z, 2.0)) * -0.4046220386999212);
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (z <= -6500000000.0)
		tmp = Float64(fma(0.0692910599291889, y, x) + Float64(Float64(y / z) * Float64(0.07512208616047561 + Float64(-0.4046220386999212 / z))));
	elseif (z <= 155000.0)
		tmp = fma(y, Float64(fma(z, Float64(cbrt(fma(z, 0.0692910599291889, 0.4917317610505968)) * cbrt((fma(z, 0.0692910599291889, 0.4917317610505968) ^ 2.0))), 0.279195317918525) / fma(z, Float64(z + 6.012459259764103), 3.350343815022304)), x);
	else
		tmp = Float64(Float64(Float64(2.181088706546648 * Float64(y / (z ^ 3.0))) + Float64(Float64(Float64(y / z) * 0.07512208616047561) + Float64(x + Float64(0.0692910599291889 * y)))) + Float64(Float64(y / (z ^ 2.0)) * -0.4046220386999212));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[z, -6500000000.0], N[(N[(0.0692910599291889 * y + x), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(0.07512208616047561 + N[(-0.4046220386999212 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 155000.0], N[(y * N[(N[(z * N[(N[Power[N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(2.181088706546648 * N[(y / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y / z), $MachinePrecision] * 0.07512208616047561), $MachinePrecision] + N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] * -0.4046220386999212), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.4
Target	0.3
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -6.5e9

   1. Initial program 42.1

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Simplified 34.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]

   3. Taylor expanded in z around inf 0.3

      \[\leadsto \color{blue}{\left(0.07512208616047561 \cdot \frac{y}{z} + \left(0.0692910599291889 \cdot y + x\right)\right) - 0.4046220386999212 \cdot \frac{y}{{z}^{2}}} \]

   4. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right) + \frac{y}{z} \cdot \left(0.07512208616047561 - \frac{0.4046220386999212}{z}\right)} \]

   if -6.5e9 < z < 155000

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]

   3. Applied egg-rr 0.1

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \color{blue}{\sqrt[3]{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)\right)}^{2}}}, 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right) \]

   if 155000 < z

   1. Initial program 41.0

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Simplified 33.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)} \]

   3. Taylor expanded in z around inf 0.3

      \[\leadsto \color{blue}{\left(2.181088706546648 \cdot \frac{y}{{z}^{3}} + \left(0.07512208616047561 \cdot \frac{y}{z} + \left(0.0692910599291889 \cdot y + x\right)\right)\right) - 0.4046220386999212 \cdot \frac{y}{{z}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6500000000:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right) + \frac{y}{z} \cdot \left(0.07512208616047561 + \frac{-0.4046220386999212}{z}\right)\\ \mathbf{elif}\;z \leq 155000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \sqrt[3]{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)\right)}^{2}}, 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2.181088706546648 \cdot \frac{y}{{z}^{3}} + \left(\frac{y}{z} \cdot 0.07512208616047561 + \left(x + 0.0692910599291889 \cdot y\right)\right)\right) + \frac{y}{{z}^{2}} \cdot -0.4046220386999212\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))